Type the equation $f\left(x\right)\=\dots$

Context Panel: Assign Function

Figure 1.2.6(a) is an animation in which $y\=6$ is graphed in blue, and $f\left(x\right)\=\frac{x^2-9}{x-3}$, in black.

The slider in the animation toolbar controls the value of $\mathrm{\ϵ}$. As the slider is moved past the first frame, red and green horizontal lines delineate an $\mathrm{\ϵ}$-band around $y\=6$ and red and green vertical lines delineate a corresponding $\mathrm{\δ}$-band around $x\=3$.

The red and green horizontal lines are drawn at $y\=6 \pm \mathrm{\ϵ}$, respectively, and the red and green vertical lines are drawn at the corresponding $x$-coordinates $x\=3 \pm \mathrm{\δ}\=f^{-1}\left(6 \pm \mathrm{\ϵ}\right)$. (Because $f$ reduces to the linear $x\+3$ for $x\ne 3$, once again ${\mathrm{\δ}}_L\={\mathrm{\δ}}_R\=\mathrm{\δ}\left(\mathrm{\ϵ}\right)$).

Write the equation $f\left(a-{\mathrm{\δ}}_L\right)\=L- \mathrm{\ϵ}$ 
Press the Enter key.

Context Panel: Simplify≻Simplify

Context Panel: Solve≻Isolate Expression for≻${\mathrm{\δ}}_L$ 

Write the equation $f\left(a\+{\mathrm{\δ}}_R\right)\=L\+\mathrm{\ϵ}$ 
Press the Enter key.

Context Panel: Simplify≻Simplify

Context Panel: Solve≻Isolate Expression for≻${\mathrm{\δ}}_R$